# Romanov's Integral

**Calculus**Level 5

This integral comes from my personal research and IMHO, it should be a level 5 problem. Why? First, neither Wolfram|Alpha nor Mathematica can return a plausible closed-form for this integral. Not only a closed-form that they failed to give, but also its numerical value to the precision of **only** 10 digits (using normal procedure). Second, it has been posted at four different sites such as: Mathematics StackExchange, Integrals and Series, AoPS, and Quora, but none exact solution has been given **yet**. So, if you could answer it, would you care to post the solution? Here is the problem:

\[\begin{equation} \large\mathscr{R}=\int_0^{\Large\frac{\pi}{2}}\sin^2x\,\ln\big(\sin^2(\tan x)\big)\,\,dx=\frac{\pi}{\alpha}\ln\left(\frac{e^\beta-\gamma}{\delta}\right)-\frac{\pi}{\mu}\left(\frac{e^\nu-\theta}{e^\lambda-\psi}\right)+\omega \end{equation}\]

where \(\large \alpha,\beta,\gamma,\delta,\mu,\nu,\theta,\lambda,\psi,\omega\) are non-negative integers and square-free. Find \(\large \alpha+\beta+\gamma+\delta+\mu+\nu+\theta+\lambda+\psi+\omega\)?